Abstract
Covariance control methods have been applied to linearstochastic multivariable control systems to ensure good behavior of eachstate variable separately. Recent attempts to extend these ideas tononlinear systems have been reported, including an example of a systemexhibiting hysteresis nonlinearity which employed describing functions.As nonlinearities, including hysteresis, occur frequently in structuralsystems, the development of effective control algorithms to accommodatethem is desirable. Recently, the authors designed covariance controllersfor several hysteretic systems using the method of stochastic equivalentlinearization. Performance of the closed loop system employing thecovariance control was verified through simulation. In the present work,a new control design method is adopted that uses the principle ofmaximum entropy, which has been used as an alternative procedure forclosure of moment equations arising in stochastic dynamical systems. Themaximum entropy-based method leads to a result equivalent to that ofstochastic linearization when covariances alone are specified; however,the method readily accommodates the specification of higher orderresponse moments.
Similar content being viewed by others
References
Hotz, A. and Skelton, R. E., ‘Covariance control theory’, International Journal of Control 46(1), 1987, 13–32.
Skelton, R. E., Iwasaki, T., and Grigoriadis, K., A Unified Algebraic Approach to Control Design, Taylor and Francis, Bristol, PA, 1997.
Skelton, R. E. and Delorenzo, M. L., ‘Weight selection for variance-constrained LQG regulators with application to the ‘hoop-column’ antenna’, in 4th VPI & SU/AIAA Symposium on Dynamics and Control of Large Space Structures, L. Meirovitch (ed.), VPI & SU, Blacksburg, 1983, pp. 69–86.
Delorenzo, M. L., ‘Sensor and actuator selection for large space structure control’, AIAA Journal of Guidance, Control, and Dynamics 13Y(2), 1990, 249–257.
Field, R. V. and Bergman, L. A., ‘Reliability-based approach to linear covariance control design’, ASCE Journal of Engineering Mechanics 124(2), 1999, 193–198.
Chung, H. Y. and Chang, W. J., ‘Extension of the covariance control principle to nonlinear stochastic systems’, IEE Proceedings: Control Theory and Applications 141(2), 1994, 93–98.
Chang, K. Y., Wang, W. J. and Chang, W. J., ‘Covariance control for stochastic multivariable systems with hysteresis nonlinearity’, International Journal of Systems Science 28(7) 1997, 731–736.
Beaman, J. J. and Hedrick, J. K., ‘Improved statistical linearization for analysis and control of nonlinear stochastic systems. Part II: Application to control system design’, ASME Journal of Dynamic Systems, Measurement, and Control 103, 1981, 114–119.
Beaman, J. J., Nonlinear quadratic Gaussian control', International Journal of Control 39(2), 1984, 343–361.
Yoshida, K., ‘A method of optimal control of non-linear stochastic systems with non-quadratic criteria’, International Journal of Control 39(2), 1984, 179–291.
Young, G. E. and Chang, R. J., ‘Optimal control of stochastic parametrically and externally excited nonlinear control systems’, ASME Journal of Dynamic Systems, Measurement, and Control 110, 1988, 114–119.
Itô, K., ‘On a formula concerning stochastic differentials’, Nagoya Math Journal 3, 1951, 55–65.
Sobczyk, K. and Trebicki, J., ‘Maximum entropy principle in stochastic dynamics’, Probabilistic Engineering Mechanics 3(5), 1990, 102–110.
Sobczyk, K. and Trebicki, J., ‘Maximum entropy principle and non-linear stochastic oscillators’, Physica A 193, 1993, 448–468.
Trebicki, J. and Sobczyk, K., ‘Maximum entropy principle and nonstationary distributions of stochastic systems’, Probabilistic Engineering Mechanics 9, 1996, 169–178.
Bergman, L. A, Wojtkiewicz, S. F., Turan, G. A., and Dyke, S. J., ‘Covariance control of a system with elasto-plastic hysteresis’, in Proceedings of the Third International Conference on Computational Stochastic Dynamics, P. Spanos (ed.), Balkema, Rotterdam, 1999, pp. 255–259.
Wojtkiewicz, S. F., Bergman, L. A., and Turan, G., ‘Covariance control of a hysteretic system’, in Proceedings of the Second World Conference on Structural Control, T. Kobori, Y. Inoue, K. Seto, H. Iemura, and A. Niskitani (eds.), Wiley, Chichester, U.K., 1999, pp. 1967–1974.
Lutes, L. D. and Sarkani, S., Stochastic Analysis of Structural and Mechanical Vibrations, Prentice-Hall, Englewood Cliffs, NJ, 1997.
Johnson, E. A., Wojtkiewicz, S. F., Bergman, L. A., and Spencer, B. F. Jr., ‘Observations with regard to massively parallel computation for Monte Carlo simulation of stochastic dynamical systems’, International Journal of Non-Linear Mechanics 32(4), 1997, 721–734.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Wojtkiewicz, S.F., Bergman, L.A. A Moment Specification Algorithm for Control of Nonlinear Systems Driven by Gaussian White Noise. Nonlinear Dynamics 24, 17–30 (2001). https://doi.org/10.1023/A:1026575320113
Issue Date:
DOI: https://doi.org/10.1023/A:1026575320113